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Mirrors > Home > MPE Home > Th. List > canthwdom | Structured version Visualization version GIF version |
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9132, equivalent to canth 7358). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
canthwdom | ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5347 | . . . . 5 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | ne0i 4329 | . . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅) | |
3 | 1, 2 | mp1i 13 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅) |
4 | brwdomn0 9566 | . . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) |
6 | 5 | ibi 267 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
7 | relwdom 9563 | . . . . 5 ⊢ Rel ≼* | |
8 | 7 | brrelex2i 5726 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V) |
9 | foeq2 6796 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥)) | |
10 | pweq 4611 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
11 | foeq3 6797 | . . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
13 | 9, 12 | bitrd 279 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
14 | 13 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴)) |
15 | vex 3472 | . . . . . 6 ⊢ 𝑥 ∈ V | |
16 | 15 | canth 7358 | . . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥 |
17 | 14, 16 | vtoclg 3537 | . . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
18 | 8, 17 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
19 | 18 | nexdv 1931 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
20 | 6, 19 | pm2.65i 193 | 1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∅c0 4317 𝒫 cpw 4597 class class class wbr 5141 –onto→wfo 6535 ≼* cwdom 9561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-wdom 9562 |
This theorem is referenced by: pwdjudom 10213 |
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